# Whole number

Whole numbers are non-negative numbers including 0 that do not have a fractional or decimal component.

## Whole number definition

Whole numbers are numbers that are not fractions, decimals, or negative. They can also be defined as a subset of the integers, or even in terms of counting numbers: Whole numbers are all the positive integers as well as 0; they can also be defined as the set of cardinal numbers, also referred to as counting numbers, including 0.

### Set of whole numbers

The set of whole numbers is commonly represented by the whole number symbol W and can be written as follows:

{0, 1, 2, 3, 4, 5, 6, ...}

### Whole numbers on a number line

Whole numbers can also be expressed using a number line:

Note that only the points on the number line that have labeled values are whole numbers. Depending on the definitions used, whole numbers may be synonymous with natural numbers.

### Smallest and largest whole number

The smallest whole number is 0. There is no largest whole number because the set of whole numbers continues towards infinity.

## Difference between whole numbers and natural numbers

The key difference between the set of whole numbers and natural numbers is that the set of whole numbers includes 0 while the set of natural numbers does not. Essentially, every natural number is a whole number and every whole number except for 0 is a natural number. We can see this when comparing the two sets. The set of whole numbers is,

{0, 1, 2, 3, 4, 5, 6, ...},

and the set of natural numbers is:

{1, 2, 3, 4, 5, 6, ...}

In some cases, the set of natural numbers is defined as including 0. Under that definition, there would not be a difference between natural numbers and whole numbers, but this definition is less common. The table below is a summary of the key differences between the sets of numbers:

Whole numbers | Natural numbers |
---|---|

Includes 0 | Doesn't include 0 |

{0, 1, 2, 3, 4, 5, 6, ...} | {1, 2, 3, 4, 5, 6, ...} |

Smallest number is 0 | Smallest number is 1 |

## Whole numbers and integers

Integers include negative values. Aside from this difference, the set of whole numbers and integers include the same values. This means that a whole number is always an integer, but an integer is not always a whole number.

The set of whole numbers: | {0, 1, 2, 3, 4, 5, 6, ...} |

The set of integers: | {..., -3, -2, -1, 0, 1, 2, 3, ...} |

If we took the absolute value of any integer, we would get a whole number.

### Can whole numbers be negative

Whole numbers cannot be negative. They are defined as the set of positive integers including 0.

## Is 0 a whole number?

Yes 0 is a whole number. This is simply by definition. It is included in the set of whole numbers as part of the definition.

## Properties of whole numbers

Below are some properties of whole numbers.

### Closure property of whole numbers

The closure property states that given two whole numbers, a and b,

a × b → whole number

a + b → whole number

In other words, if we add or multiply two whole numbers, the result will be a whole number.

### Commutative property of whole numbers

The commutative property states that the order in which we add or multiply whole numbers doesn't matter; the result will be the same. Given two whole numbers, a and b,

a × b = b × a

a + b = b + a

### Additive identity of whole numbers

The additive identity is 0. If 0 is added to any whole number, the result is the same whole number. Given a whole number a,

a + 0 = 0 + a

Thus, 0 is referred to as the additive identity of whole numbers.

### Multiplicative identity of whole numbers

The multiplicative identity of whole numbers is 1. If 1 is multiplied by any whole number, the result is the same whole number. Given a whole number a,

a × 1 = a

Thus, 1 is referred to as the multiplicative identity of whole numbers.

### Associative property of whole numbers

The associative property of whole numbers states that whole numbers being added or multiplied can be grouped in any order; the result will be the same. Given three whole numbers a, b, and c,

(a × b) × c = a × (b × c)

(a + b) + c = a + (b + c)

### Distributive property of whole numbers

The distributive property states that multiplication of a whole number is distributed over the sum or difference of the whole numbers. Given three whole numbers a, b, and c,

a × (b + c) = (a × b) + (a × c)